The Nobel Prize in Physiology or Medicine 2014

The Nobel Prizein Physiology or Medicine 2014 is awarded for discovering how the brain navigates our way – our inner “GPS” system in the brain. The prize goes to J. O’Keefe (UCL, UK) and E. Moser and M.B. Moser (University of Trondheim, Norway).

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MEDICINE APPLIED AND CHEMISTRY MODEL IN LAGRANGIAN
ABSTRACT
Those are the Hamiltonian equations of motion. Instead of a single second-order equation for each coordinate, we have two first-order equations, which may be easier to solve.

The derivation I gave above was hardly air-tight. However, it’s easy to verify that in Newtonian mechanics using Cartesian coordinates, the Hamiltonian we obtain from equation (h.6) reduces to:
(h.8)

which is just the total energy. In that case it’s also easy to verify equations (h.7) directly. As with the Lagrangian formulation, however, much of the value of the Hamiltonian formulation lies in the fact that equations (h.7) are true regardless of the coordinates we’re using. Also keep in mind that equation (h.8) is only necessarily true when the Lagrangian chosen is the “pure Newtonian” T-V. For other Lagrangians, the Hamiltonian won’t necessarily be the total energy of the system.otion

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CHEMISTRY AND MEDICINE AND LAGRANGIAN OPERATOR
ABSTRACT

To find out the dependence of pressure on equilibrium temperature when two phases coexist.

Along a phase transition line, the pressure and temperature are not independent of each other, since the system is univariant, that is, only one intensive parameter can be varied independently.When the system is in a state of equilibrium, i.e., thermal, mechanical and chemical equilibrium, the temperature of the two phases has to be identical, the pressure of the two phases has to be equal and the chemical potential also should be the same in both the phases.
INTRODUCTION

Consider a system consisting of a liquid phase at state 1 and a vapour phase at state 1’ in a state of equilibrium. Let the temperature of the system is changed from T1 to T2 along the vaporization curve.

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USING OF ELLIPTICAL METHODS IN MEDICINE AND CHEMISTRY
ABSTRACT

The J0 Bessel function
The equation for J0 Bessel is the zeroth order Bessel equation
.
The “standard form” of differential equations is often specified as having the coefficient of the highest order derivative cancelled through. Thus in standard form the equation would be written
.

Our procedure for the series solution of this equation is to take the assumed series

and substitute it into the equation. This involves using the first and second derivatives

and

In the expression for the second derivative we have (as in the sine/cosine case above) shifted the dummy summation variable by 2 so that the sum expression contains xn explicitly.

So far we have left the sum for the first derivative unchanged. The point here is that what the differential equation contains is , and the expression for this must be written as a sum in xn. For this reason we shift the dummy variable in the series for the first derivative by 2:

so that
.
We now substitute the series expressions into the original differential equation:
.
The term in the second sum may be treated separately. In that case everything else falls within a sum over n from 0 to .
.
As we argued in the previous example, this expression is valid for all values of the independent variable x, so that each power of x must vanish separately.

Look at the –1 term first. The requirement that this term vanish means that
.
Now look at the general case:
.
This may be tidied into
,
which gives a recurrence relation for the coefficients:
.
We may now build up the coefficients from the term. Starting from we find
.
Now putting gives
1 INTRODUCTION
The J0(x) function goes to 1 as x goes to 0. This we see on the plot and we have discovered this in the series solution. The Y0(x) function, on the other hand, looks as if it is heading for minus infinity as x goes to 0. That is the problem.

Recall the point made when we introduced the power series method. A series

will only work when the function is “well behaved”. This is OK for J0(x), but going off to infinity is an example of “bad behaviour”; then a simple power series won’t work. We will see how to overcome this in a later section.

The important concepts of this section are:

• The simple power series method works only for “well-behaved” functions; it cannot cope with “badly-behaved” functions.

• The basic idea is to substitute the power series into the differential equation.

• With a slight juggling of the dummy summation variables the equation is cast into the form . Special care must be taken with the first derivative term.

• Each power of x must equate to zero.

• The term in x-1 must be treated separately; this tells us that there is no simple series in odd powers of x. The series method is only going to give us one solution to the ODE, which is even in x.

• Equating the general term in xn to zero gives a recurrence relation for the coefficients.

• We build up one solution to the ODE from the a0 coefficient.

• A 2nd order ODE has two independent solutions; where is the other? We recognise that the simple power series method can’t cope with it as it is “badly-behaved”.

• Properties of the J0 function are obtained from the series solutions and the original ODE.

Legendre’s equation
Legendre’s equation follows from separating the laplacian in spherical polar coordinates. This equation arises from the separated equation in the polar angle . Legendre’s equation is
.
In this equation n is often a positive integer; we will explore this a little later.

As before, we start with a power series expression for the function

Now, however, we will use Mathematica to obtain the recurrence relation for the coefficients. This is outlined in the Mathematica Notebook “Legendre”. The recurrence relation is
.

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METHODS IN MEDICINE AND CHEMISTRY WITH LAGRANGIAN
ABSTRACT

The J0 Bessel function
The equation for J0 Bessel is the zeroth order Bessel equation
.
The “standard form” of differential equations is often specified as having the coefficient of the highest order derivative cancelled through. Thus in standard form the equation would be written
.

Our procedure for the series solution of this equation is to take the assumed series

and substitute it into the equation. This involves using the first and second derivatives

and

In the expression for the second derivative we have (as in the sine/cosine case above) shifted the dummy summation variable by 2 so that the sum expression contains xn explicitly.

So far we have left the sum for the first derivative unchanged. The point here is that what the differential equation contains is , and the expression for this must be written as a sum in xn. For this reason we shift the dummy variable in the series for the first derivative by 2:

so that
.
We now substitute the series expressions into the original differential equation:
.
The term in the second sum may be treated separately. In that case everything else falls within a sum over n from 0 to .
1 INTRODUCTION
So the solution to the ODE which we have discovered is a constant times the J0 Bessel function
.
Thus far this is quite good; we have discovered a new function which solves the above differential equation. But it is a second order differential equation and therefore, as with the previous SHO equation, there should be two independent solutions. Where is the other solution?

When we examined the solution of the wave equation for a drumhead we found the separated radial equation took the form of the zeroth order Bessel equation. And at that stage we simply noted that Mathematica gave, as independent solutions to that equation, the two zeroth order Bessel functions J0(x) and Y0(x). We plotted the functions and the behaviour of the functions in the vicinity of x = 0 gives us an important clue about the “other” solution.

J0(x) and Y0(x) Bessel functions

The J0(x) function goes to 1 as x goes to 0. This we see on the plot and we have discovered this in the series solution. The Y0(x) function, on the other hand, looks as if it is heading for minus infinity as x goes to 0. That is the problem.

Recall the point made when we introduced the power series method. A series

will only work when the function is “well behaved”. This is OK for J0(x), but going off to infinity is an example of “bad behaviour”; then a simple power series won’t work. We will see how to overcome this in a later section.

The important concepts of this section are:

• The simple power series method works only for “well-behaved” functions; it cannot cope with “badly-behaved” functions.

• The basic idea is to substitute the power series into the differential equation.

• With a slight juggling of the dummy summation variables the equation is cast into the form . Special care must be taken with the first derivative term.

• Each power of x must equate to zero.

• The term in x-1 must be treated separately; this tells us that there is no simple series in odd powers of x. The series method is only going to give us one solution to the ODE, which is even in x.

• Equating the general term in xn to zero gives a recurrence relation for the coefficients.

• We build up one solution to the ODE from the a0 coefficient.

• A 2nd order ODE has two independent solutions; where is the other? We recognise that the simple power series method can’t cope with it as it is “badly-behaved”.

• Properties of the J0 function are obtained from the series solutions and the original ODE.

Legendre’s equation
Legendre’s equation follows from separating the laplacian in spherical polar coordinates. This equation arises from the separated equation in the polar angle . Legendre’s equation is
.
In this equation n is often a positive integer; we will explore this a little later.

As before, we start with a power series expression for the function

Now, however, we will use Mathematica to obtain the recurrence relation for the coefficients. This is outlined in the Mathematica Notebook “Legendre”. The recurrence relation is
.

A series in even powers of x will be built up from a0 and a series in odd powers of x will be built up from odd powers of x. The general solution to the Legendre equation is thus

where

Often n is a positive integer. In that case: if n is even then the series for will terminate at the xn term, while if n is odd then the series for will terminate at the xn term. These solutions, normalised to , are called the Legendre polynomials, denoted by . The first few are given by

They are plotted in the following figure

First few Legendre polynomials

When n is an integer one series solution of the Legendre terminates and we thus have the Legendre polynomials The other series solution does not terminate. These are denoted by , and they can be expressed in terms of logarithms:

They are plotted in the following figure

First few Legendre Qn functions

The general solution of the Legendre equation will be
.
But note that has a logarithmic divergence at ; the only well-behaved solutions of Legendre’s equation for integer n are the Legendre polynomials

The important concepts of this section are:

• Mathematica can be used to derive the coefficient recurrence relation by substituting some general terms of the power series into the differential equation.

• The recurrence relation connects every other coefficient.

• Therefore there are two independent solutions, one in the even powers of x and one in the odd powers.

• For the Legendre equation with integer n the coefficients of one of the series solutions will terminate.

• If n is even then the series for will terminate at the xn term, while if n is odd then the series for will terminate at the xn term.

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FUBINI THEOREM , FATOU CONSEQUENCES INTEGRATION IN CHEMISTRY
ABSTRACT
One proves theorems such as if is a refinement of partition then and (that is, as you make the refinement finer…with smaller intervals, the lower sums go up (or stay the same) and the upper sums go down (or stay the same) and then one can define to thebe infimum (greatest lower bound) of all of the possible upper sums and to the the supremum (least upper bound) of all of the possible lower sums. If we then declare that to be the (Riemann) integral of over
Note that this puts some restrictions on functions that can be integrated; for example being unbounded, say from above, on a finite interval will prevent upper sums from being finite. Or, if there is some dense subset of for which obtains values that are a set distance away from the the values that attains on the compliment of that subset, the upper and lower sums will never converge to a single value. So this not only puts restrictions on which functions have a Riemann integral, but it also precludes some “reasonable sounding” convergence theorems from being true.

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CHEMISTRY IN CONNECTION WITH MATHEMATICS OPERATORS
ABSTRACT
Velocity Potentials and Stream Functions
As we have seen, a two-dimensional velocity field in which the flow is everywhere parallel to the – plane, and there is no variation along the -direction, takes the form
(5.16)

Moreover, if the flow is irrotational then is automatically satisfied by writing , where is termed the velocity potential. (See Section 4.15.) Hence,
(5.17)
(5.18)

On the other hand, if the flow is incompressible then is automatically satisfied by writing , where is termed the stream function. (See Section 5.2.) Hence,
(5.19)
(5.20)

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that
(5.21)
(5.22)

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FROBENIUS CONCEPTS IN INTEGRATION AND APPLICATION. FOR FACULTIES Chemistry and Pharmaceutic Bucharest
ABSTRACT
In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat’s theorem.

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LAGRANGIAN AND HURWITZ CONSIDERATIONS
ABSTRACT
The following problems involve the use of l’Hopital’s Rule. It is used to circumvent the common indeterminate forms “0”0 and “∞”∞
when computing limits. There are numerous forms of l”Hopital’s Rule, whose verifications require advanced techniques in calculus, but which can be found in many calculus books. This link will show you the plausibility of l’Hopital’s Rule. Following are two of the forms of l’Hopital’s Rule.
THEOREM 1 (l’Hopital’s Rule for zero over zero): Suppose that limx→af(x)=0
, limx→ag(x)=0, and that functions f and g are differentiable on an open interval I containing

It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets and thus are considered results of emerich toth , hangan that have descovered line, points as marcel chirita, , ene horia , andrei bernescu , or sorin radulescu with cristian alexandrescu these must lead a other

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